Prime Numbers

584

fundamental theorem (of arithmetic), 1, 2, 6, 7, 34, 52

Furry, W., 65

G¨ohl, G., 171

Gabcke, W., 67 Gage, P., 23

Gallot, Y., 14, 30, 31, 63, 456, 528 Galois

group, 191, 299, 359 theory, 152

Galway functions, 161, 168 Galway, W., x, 68, 161, 170 Gandhi formula, 50 Gardner, M., 2

Garner algorithm, 88 Garst, B., x, 395 Gaudry, P., 357 Gauss

quadratic reciprocity, 111 sums, 99, 111, 194–196, 199,

211, 221, 526

Gauss, C., 10, 11, 14, 27, 44, 100, 115, 210, 239, 241, 404, 478

Gaussian

elimination, 268, 269 integers, 258, 282, 306, 500 matrix methods, 304 period, 207, 210, 212 random variable, 426 rationals, 306

unitary ensemble (GUE), 426 Gazzoni, D., 79

generalized Riemann hypothesis (GRH), 26, 42, 113, 114, 207, 223

Gentleman, W., 483 Gerhard, J., 95 Gerlach, H., 165 Gesley, M., x

Glasserman, P. (with Boyle et al.), 413

Goldbach conjecture, 18–20, 46, 48, 62, 71, 72, 491, 492, 536

Goldbach, C., 18

INDEX

golden mean, 430, 431, 529 Goldfeld, D, 247 Goldston, D, 38

Goldwasser, S., 368, 371, 397 Goldwasser–Kilian, 373 Golomb, S., 162, 400

Gong, G., x, 402, 438 Gordon, D., 81, 149, 306 Gourdon, X., 11, 12, 16, 38

Govaerts, R. (with Bosselaers et al.), 449, 452, 453

Gra , M., 3

Graham, S., 45, 59, 111, 113 grammar-school methods, 109, 342,

378, 443–445, 453, 473, 474, 503, 505, 506, 519, 522

Grantham, J., 145, 149–152, 168 Granville identity (combinatorial),

82

Granville, A., x, 38, 49, 134, 166, 209, 417

Granville, A. (with Alford et al.), 134, 135, 141

Gray code, 275, 276, 310

greatest common divisor (gcd), 83 binary, 110, 463–466, 523 Euclid, 84, 513

extended binary, 464 extended for inverse, 85, 89,

106, 108

fast-binary recursive, 470 polynomial, 89–91, 95, 512 recursive, 466, 468, 471, 522

Green, B., ix, 13, 14 Gri ths, D., x Gross, B., 247 ground states, 421

group automorphism, 352 Guy, R., 7, 77, 231, 416

H˚astad, J., 433

Hadamard, J., 10

Hafner, J., 251

Hagis, P., 25

Haglund, J., 165

INDEX

Hajratwala, N., 23 Halberstam, H., 16, 18, 19

Halton sequence, 408–410, 413, 414, 433

Hamiltonian, 427 Hammersley point set, 409 Hardy function, 66

Hardy, G., 36, 46, 48, 87, 121, 227, 492, 527, 538

Hardy–Krause bounded variation, 406

Hardy–Littlewood convexity conjecture, 20

Harley, R., x, 358 Harman, G., 37 Hartley transform, 478 hash function, 390 Hasibar, E., x

Hasse theorem, 333, 334, 337, 339, 349, 350, 368, 378

Hasse, H., 333, 334, 350 Hayes, D., x Heath-Brown, D., 45, 142

Heaviside function, 158, 412 Hellman, M., 387

Hensel lifting, 105, 292, 304 Hensel, K., 105

Hensley, D., 20, 81 Hensley-Richards result, 81 Hermite polynomials, 427, 428 Hermitian operator, 426 Herranen, K., 30

Hey, T., 420 high 5 jive, 430

Higham, N., 436, 498 Hilbert

class field, 359

class polynomial, 359–364, 366 space, 421

tenth problem, 21, 417, 418 Hilbert, D., 359

Hilbert–P´olya conjecture, 426 Hildebrand, A., 114

Hill, D., x

Hippocratic healing, 430 Hofmann, U., x

585

Holdridge, D., 273 holography, 419 Honaker, G., 79

Hooley, C., 77, 113, 207, 223 Hopcroft, J. (with Aho et al.), 466,

509, 513 Horner’s rule, 518 Howgrave-Graham, N., x

Howgrave-Graham, N. (with Coppersmith et al.), 189

Huang, J., x Huang, M., 200, 371 Hudson, R., 59 Hung C., 453 Hurwitz, A., 29 Hutchinson, D., 428

hypergeometric series, 381 hyperspherical group, 258 hypothesis H, 17, 18

imaginary quadratic fields, 359 Impagliazzo, R. (with H˚astad et

al.), 433 Indlekofer, K.-H., 14 industrial-grade prime, 138 Intel, 17

inverse discrete Fourier transform, 505

inverse fast Fourier transform, 482, 485, 502, 538

irrational-base discrete weighted transform (IBDWT), 23, 496, 497, 528

Ivi´c, A., 36, 69 Iwaniec, H., 18, 45 Izu, T., 357

J. Arney, 251

J. Selfridge, 79 J´arai, A., 14

Jacobi sums, 194, 215

Jacobi symbol, 40, 97, 98, 143, 166, 182, 324, 368, 382

Jacobi, C., 115

Jacobian varieties (of hyperelliptic curves), 371

586

Jacobson Jr., M. (with Buchmann et al.), 236

Jaeschke, G., 169, 222 Jarvis, N., 81

Jobling, P. (with Frind et al.), 78 Jobs, S., x

Joe, S., 415

Johnson, D., 394, 523 Jones, A., x

Jones, E., 488

Jones, J., 418

Jozsa, R., 420

Jullien, G. (with Dimitrov et al.), 530

Juriˇsi´c, A., 394

Kaczorowski, J., 59

Kaliski Jr., B., x, 375, 376, 395 Kaliski Jr., B. (with Ko¸c et al.),

449, 453

Karatsuba method, 472–474, 476, 503, 505, 506, 524, 525

Karatsuba, A., 454, 473

Kayal, N., 200, 212, 217

Kayal, N. (with Agrawal et al.), 201, 207

Keller, J. (with Thomas et al.), 465 Keller, W., x, 30, 64, 521 Kerchner III, C., 63

Kida, M., x Kilian, J., 368, 371 Kim, K., x

Kiss, P. (with Erd˝os et al.), 149 Kleinjung, T., 4

Kleinjung, T. (with Franke et al.), 374

Klivington, J., x, 488

Knuth, D., 85, 163, 447, 453, 463, 464, 466, 476, 506

Ko¸c C., 449, 453 Koblik, K., x Koblik, S., x

Koblitz, N., 321, 391, 394, 395, 432 Kocis, L., 414

Kogure, J. (with Izu et al.), 357 Kohel, D., x

INDEX

Kolesnik, G., 45, 59, 111 Konyagin, S., 49, 176 Korobov, N., 70

Korselt criterion, 164 Korselt, A., 133, 134 Kramer, D., x, 488 Kraus, A., 417 Kronecker

class number, 334 symbol, 247

Kruppa, A., x, 30 Kuipers, L., 58, 439 Kummer, E., 211, 492 Kurlberg, P., 401 Kurowski, S., x, 23

Lagarias, J., 152, 155, 158, 159, 161, 401, 402, 433, 441

Lagarias–Odlyzko identity, 160 Lagrange descent, 527

Lagrange theorem, 191, 435, 527 Lam´e theorem, 84, 108

Lanczos method, 269 Landau, E., 80 Landau, S., x

Landau–Ramanujan constant, 80 Landrock, P. (with Damg˚ard et

al.), 138, 164 Langevin, M., 416 Languasco, A., 72

large-integer multiplication, 473 Larsen, G. (with Thomas et al.),

465 Lavenier, D., 536

least common multiple (lcm ), 108 Lee, P., 391

Legendre polynomial, 381 relation, 51

symbol, 96–99, 113–115, 142, 143, 152, 174, 182, 247, 265, 289, 291, 333, 334, 381, 402, 434

Legendre, A., 11, 52, 115, 239, 404 Lehman, R., 227, 228

Lehmer method (for gcd), 463

INDEX

Lehmer, D., 149, 152, 155, 164, 166, 174, 175, 186, 463, 464, 466

Lehmer, D. (with Brillhart et al.), 299

Lehmer, E., 164

Lenstra Jr., H., x, 31, 57, 76, 107, 128, 166, 186, 187, 189– 191, 194, 207, 209–212, 238, 251, 278, 302, 323, 333–336, 339

Lenstra Jr., H. (with Buhler et al.), 289, 290, 292, 297, 493, 532

Lenstra Jr., H. (with Lenstra et al.), 28, 280, 293, 299

Lenstra, A., x, 3, 28, 189, 272, 278, 280, 292, 293, 299, 301, 302

Leveque theorem, 439 Levich, M., x

Levin, L. (with H˚astad et al.), 433 Levy flight, 428

Leyland, P., 3, 4, 374 lg, 26

li, 10 li0, 59

Li, S., 132, 169 Li, X., 441

Lichtblau, D., x, 311 Lieman, D., x Lifchitz, H., 14

Lim, C., 391

Lindel¨of hypothesis, 45 Lindemann, I., x Lindqvist, L., 80 Littlewood, J., 46, 59, 78 Liu, Q., 487 Loebenberger, D., x Long, D., 389

Lovasz, L., 189

Lovasz, L. (with Lenstra et al.), 280, 293

Lovorn Bender, R., 306

Luby, M. (with H˚astad et al.), 433 Lucas

587

chain, 147, 330, 345 numbers, 103, 112, 195 pseudoprimes, 142, 145, 146,

148–150, 166 theorem, 173, 174, 179 tree, 179, 181

Lucas, E., 28, 173, 174, 179 Lucas–Lehmer test, 4, 23, 24, 76,

181, 183, 184, 456, 496, 523, 528

Lyapunov exponents, 438 Lygeros, N. (with Dubner et al.),

78, 79 Lyne, A., 483

M¨uller, V., x, 392 Madisetti, V., 499 Maier, H., 38 Mairson, 126 Manasse, M., 272

Manasse, M. (with Lenstra et al.), 28, 299

Mann, X., 433

Marcus, D., 284 Marsaglia theorem, 401 Marsaglia, G., 400, 438 Martin, M., x

Masser, D., 417

Mathematica, viii, 311, 541, 543 Mathematical Games, 2 Matijaseviˇc, Y., 418

matrix, 151, 304

and random numbers, 400 determinant, 239

fast methods, 259, 265, 268 GUE, 427

Hessenberg, 427

in convolution, 538 in FFT, 483, 484

in golden-section transform, 529

in NFS, 293

in QS, 266

in recursive gcd, 469 operators, 421 random, 426

588

reduction, 263

structured Gauss methods, 269 transpose, 485, 530

Mauldin, R., 417

Mayer, E., x, 4, 23, 30, 218, 485, 488, 538

Mayer, E. (with Crandall et al.), 30, 219, 456, 485, 491, 496, 528, 531, 538

Mayer, E. (with Crandall et al.), 218

McClellan, J., 524

McCurley, K., 251

McGuckin, F., x McIntosh, R., 28, 32, 386

McIntosh–Tardif factor, 256, 385 McIntosh–Wagsta probable prime,

384 McKee, J., 242, 255

measure theory (of continued fractions), 85

Meissel, E., 152, 155

Mellin transform, 160–162, 168 Menezes, A., 88, 91, 102, 104, 110,

390–392, 394, 402, 445, 450, 461, 522

Merel, L., 417 Mersenne

numbers, 4, 22, 25–28, 52, 63, 64, 76, 79, 109, 181, 183, 256, 307, 308, 455, 496– 498, 523

primes, 3, 22–25, 31, 52, 63, 64, 76, 79, 113, 183, 184, 363, 373, 399, 438, 454– 456, 466, 496, 500–502, 524, 528, 530–532, 539

Mersenne, M., 25 Mertens

conjecture, 39 constant, 35, 36, 79, 80 function, 36, 37

theorem, 27, 35, 127, 163 Mestre theorems, 378 Mestre, J., 349

Mets¨ankyl¨a, T., 416

INDEX

Micali scenario, 397 Micali, S., 397 Mignotte, M., x, 416

Mih˘ailescu, P., ix, x, 214, 215, 416 Miller, G., 137, 141, 200

Miller, K., 399

Miller, V., x, 152, 391, 402

Miller, V. (with Lagarias et al.), 152, 155, 158

Miller, W.(with Dimitrov et al.), 530

Mills constant, 75 Mills, W., 21, 75

Mister, S. (with De Win et al.), 381, 453, 458, 536

Mitchell, D., x

Mitchell, V., x Mitra, T., x

Miyaji, A. (with Cohen et al.), 333, 460

Mizony, M. (with Dubner et al.), 78, 79

M¨obius function µ, 36, 37, 39, 94, 155, 425

Moenck, R., 513 mole rule, 5 Monico, C., 394

Monier, L., 136, 151, 165 Monier–Rabin theorem, 151 Monte Carlo, 229, 397, 404, 405,

409, 412, 413 Montgomery

coordinates, 324, 370 curve, 374

method (for div/mod), 447 parameterization, 330, 370, 432 powering, 448

Montgomery, H., 43, 426 Montgomery, H. (with Niven et

al.), 282

Montgomery, P., x, 147, 238, 254, 256, 270, 273, 292, 299, 317, 321, 329, 330, 332, 340–343, 345, 346, 374, 375, 380, 383, 384, 395,

INDEX

432, 447–450, 453, 458, 500, 520, 521

Moore, W., x

Morain, F., 4, 77, 356–361, 368, 371–374, 376, 382

Morain, F. (with Couveignes et al.), 356

Morain, F. (with Franke et al., 374 Moran, A. (with Pritchard et al.),

78

Morrison theorem, 182 Morrison, M., 28, 261, 307 MOV threshold, 392 Murphy, B., 297

Nagaraj, S., 142

Nagaraj, S. (with Coppersmith et al.), 189

Namba, M., 375 Narkiewicz, W., 248 Nathanson, M., 48, 63, 538 Nebe, G., x

Nelson, H., 23 Neubauer, G., 39

new Mersenne conjecture, 25, 26, 79

Newman, M., 7 Newton

higher-rooting, 202, 220 method, 202, 220, 448, 450,

452, 454, 511, 512, 520, 521

reciprocation, 521 square-rooting, 107, 176, 178,

189, 202, 220, 521–523 Nguyen, N., 487

Nguyen, P., 292

Nicely, T., 15, 16

Niederreiter, H., 58, 406, 407, 409, 414, 439

Niven, I., 282

Noro, M. (with Izu et al.), 357 Norrie, C. (with Crandall et al.),

29, 456, 496, 528 Novarese, P., 23

Nowak, M., 23

589

Nowakowski, R., 7 number-theoretical transform meth-

ods, 498

Nussbaumer, H., 504–507, 532, 533 Nyman, B., 38

Odlyzko, A., x, 38, 39, 59, 67, 68, 77, 152, 158–161, 235, 255, 269, 270, 426, 427

Odlyzko, A. (with Brillhart et al.), 280, 310

Odlyzko, A. (with Lagarias et al.), 152, 155, 158

Okamoto, E., 382

Okamoto, T. (with Menezes et al.), 392

Okeya, K., 433 Oki, H., x, 522 Olivier, M., 53

Ono, T. (with Cohen et al.), 333, 460

optimal extension field (OEF), 456 Orem, F., x

Owen, A., 414

Padma, R., 367

Papadopoulos, J., x, 30, 217, 218, 481

Papadopoulos, J. (with Crandall et al.), 30, 218, 219, 456, 485, 491, 496, 528, 531, 538

Papadopoulos, P., 485

Papageorgiu, A., 413 Papp, D., 14 Parberry, E., 149 Park, S., 399

Parnami, J. (with Rishi et al.), 367 Paskov, S., 413

Patel, S., 401

Patson, N., x

Paulos, J., 430

Paun, G., 424

Peetre, J., 80

Pei, D. (with Ding et al.), 87 Penk, M., 464

590

Pentium computer chip, 17 Pepin test, 174, 218, 528 Pepin, T., 174

Peralta, R., 276, 382

Percival, C., 218, 488, 498, 528 Perez, A., x

perfect number, 24, 25, 56, 57 Perron formula, 158 Peterson, I., 5 Piatetski-Shapiro theorem, 59

Pila, J. (with Lenstra et al.), 302 Pinch, R., 134

Pintz, J., 37, 38 Pixar, 5

Pocklington theorem, 175, 213, 368 point at infinity, 320

Pollard

iteration, 251, 255

rho method, 230, 231, 233, 251, 256–258, 272, 304, 441

rho method (parallel), 235 sequence, 256, 257

Pollard, J., x, 28, 128, 229–233, 235, 236, 238, 251, 256, 259, 278, 431

Pollard, J. (with Lenstra et al.), 28, 299

P´olya–Vinogradov inequality, 114, 115

polylogarithm, 163 polynomial

arithmetic, 89, 509

division (in elliptic algebra), 352, 354

evaluation, 238, 301, 340, 343, 346, 514

inversion, 511, 513 irreducible, 92, 94, 95, 109,

192, 280 monic, 94, 296, 354

multiplication, 510, 511 remaindering, 91, 511 ring, 89

root finding, 103

INDEX

time, 80, 315, 351, 357, 368, 387, 422, 424, 436

Pomerance, C., 7, 26, 31, 49, 131, 134, 135, 149, 165, 168, 181, 194, 199, 207, 209, 210, 212, 261, 269, 273, 276, 286, 302, 305, 385, 401

Pomerance, C. (with Adleman et al.), 193, 194, 200

Pomerance, C. (with Alford et al.), 134, 135, 141

Pomerance, C. (with Buhler et al.), 289, 290, 292, 297, 493, 532

Pomerance, C. (with Canfield et al.), 49

Pomerance, C. (with Crandall et al.), 31, 33, 82, 200, 381, 534

Pomerance, C. (with Damg˚ard et al.), 138, 164

Pomerance, C. (with Friedlander et al.), 401

Pomerance, C. (with Lenstra et al.), 302

Poonen, B., 417 Powell, A., x Powell, J., x Powell, L., x

powering ladders, 103, 256, 402, 432, 448, 458, 460, 523, 535

and CRT, 109 binary, 102, 458 compressing, 458 fixed-x, 458, 462 Lucas chains, 458 Montgomery, 332 recursive, 86 sliding-window, 461 squaring, 316 ternary, 328, 523

windowing, 458, 460, 461 Prachar, K., 61

Pratt, V., 179, 180

INDEX

Preneel, B. (with De Win et al.), 381, 453, 458, 536

Preskill, J., 420

Press, W., 399, 402, 403 primality certificate, 373 primality testing, 106, 370

Adleman, Pomerance, Rumely (APR), 194

Agrawal, Kayal, Saxena (AKS), ix, 200, 202

Atkin–Morain, 131, 358, 362– 364, 371–373

elliptic curve (ECPP), 4, 323, 334, 368, 372, 380, 383, 384

elliptic curve (fastECPP), 373 finite fields, 190

Frobenius, 145, 146, 148, 150, 152

Gauss sums, 194, 196, 199, 221 Goldwasser–Kilian, 368, 370,

371, 373 Grantham-Frobenius, 145 Jacobi sums, 194, 199, 200 Lucas, 145, 146, 148, 150 Lucas–Lehmer, 4, 23, 24, 76,

181, 183, 184, 456, 496, 523, 528

McKee, 244

Miller, 142 Miller–Rabin, 137 partial factorization, 174

Pepin, 28–30, 173, 174, 181, 218–220, 380, 456

proving, 173, 324, 358, 380, 418

rapid, 385 Solovay–Strassen, 166 trial division, 120

prime k-tuples conjecture, 17, 20, 25, 63, 80, 81

prime counting, 152 alternative methods, 171 analytic method, 152, 158 combinatorial method, 152

prime life cycles, 425

591

prime number theorem (PNT), 10, 11, 34–39, 43, 51, 53, 56, 58, 61, 65, 66, 68, 108, 121, 127, 134, 138, 152

for computer scientists, 56 for residue classes, 13, 42

prime numbers in music theory, 430

prime-producing formulae, 21, 50 primes in arithmetic progression,

ix, 4, 13, 61, 63, 71, 78 primitive polynomial, 403 primitive root, 40, 93, 207, 210,

215, 223 primorial, 50 principal character, 40

principal root, 110, 115 Pritchard, P., 78, 126

probable prime, 76, 132, 135–138, 142, 148, 149, 198, 223, 369, 370, 372, 383, 384

product tree, 129, 130 projective coordinates, 324 Proth form, 24, 220, 456, 457 Proth, F., 174, 220

pseudocurves, 323, 325, 336, 337, 368

pseudoprime, 64, 131–133, 135, 136, 143, 150, 164, 166, 168, 170, 521

pseudoprime testing, 146, 173 Purdom, P., 251 Pustyl’nikov, L., 440 Putnam, H., 417, 418

quadratic Diophantine representations, 106

quadratic forms, 239, 242 quadratic nonresidues, 101, 102,

112–114, 209 quadratic residues, 96, 115, 276 quantum computation, 418

quantum fast Fourier transform (FFT), 423

quantum oscillator, 420

592

quantum Turing machine (QTM), 419, 420, 422–424, 435, 436, 441, 442

quasi-Monte Carlo (qMC), 404– 415, 433, 439

quaternion (in hypercomplex multiplication), 258

Quisquater, J.-J., 234

Rabin, M., 136, 137, 151, 200 Rader, C., 524, 532

Rajwade, A. (with Rishi et al.), 367

Ramanujan sum, 46, 57, 71, 492, 525

Ramanujan, V., 80, 166 Ramar´e, O., 20, 42

random walk, 37, 39, 113, 114, 231, 334

random-number generators, 397, 398, 400, 438

bit-wise, 401, 402 cellular-automata-based, 402 Di e–Hellman, 401 Fibonacci, 438 Golomb-Marsaglia, 400

Gong–Berson–Stinson, 402, 438 linear congruential, 398, 399,

402 Miller-elliptic, 402

multiplicative congruential, 398 power generator, 401 shift-elliptic, 402

Wu, 399 Rankin, R., 38

remainder tree, 130, 517, 534 Renze, J., x, 189

Ribenboim, P., x, 4, 19, 25, 39, 53, 75, 182, 416–418

Richards, I., 81 Richert, H., 16, 18, 19 Richstein, J., 19, 536 Riemann

conjecture, 66 critical zeros, 69, 427

INDEX

hypothesis (RH), ix, 12, 36– 38, 42, 45, 60, 61, 65, 66, 68, 69, 427, 428, 439–441

zeta function, 9, 33, 34, 36, 39, 43–45, 65, 66, 69, 80, 155, 158, 168–170, 420, 426– 429

Riemann, B., 33, 36, 37, 171 Riemann–Siegel formula, 66, 67,

159, 160 Riesel, H., 171 Ringrose, C., 113 Rishi, D., 367

risk management, 413 Rivat, J., 11, 68, 152, 158 Rivest, R., 234, 389 Robinson, J., 418 Robinson, R., 23, 166 Rodemich, G., 81 Rokhlin, V., 485, 487, 530 Rose, H., 241, 245

Rosser, J., 39, 163, 205, 218 Rotkiewicz, A., 149, 150

rounding (round()), 360, 361, 491, 496, 498

Rozenberg, G. (with Paun et al.), 424

RSA

challenge, 297 cryptosystem, 2, 389 encryption, 390, 394

RSA129 challenge number, 2, 3, 272

signature, 390, 391, 431 Rumely, R., 20, 42, 194, 199 Rumely, R. (with Adleman et al.),

193, 194, 200 running time, 9 Ruzsa, I., 82

S´ark¨ozy, A. (with Erd˝os et al.), 149 Sager, J. (with Escott et al.), 234 Saias, E. (with Balazard et al.),

440 Sakurai, K., 433

Salomaa, A. (with Ding et al.), 87

INDEX

Salomaa, A. (with Paun et al.), 424 Salzberg, B., x

Sande, G., 483 Saouter, Y., 20, 62, 536

Saouter, Y. (with Deshouillers et al.), 19

Sato, D., 418 Satoh, T., 358, 392

Saxena, N., 200, 212, 217

Saxena, N. (with Agrawal et al.), 201, 207

Sch¨onhage, A., 38, 68, 160, 466, 490, 502, 503, 507, 532

Sch¨onhage–Strassen algorithm, 507 Schaefer, E., 417

Schinzel, A., x, 17 Schirokauer, O., 306, 387 Schmid, L., 80

Schmidt, W., 407

Schneier, B., 388, 390, 397, 433 Schoenfeld, L., 39, 60, 163, 205, 218 Schoof algorithm, 351, 354, 357,

372, 378, 379, 385, 437, 533

Schoof, R., 101, 107, 199, 246, 250, 255, 349, 351, 352, 356– 358, 378, 382, 383, 437

Schoof–Elkies–Atkin (SEA) variant, 356, 357, 378, 382, 383

Schr¨odinger equation, 420 Schroeder, M., 397, 424, 425 Schulmeiss, T., x

Schur, I., 247

Scientific American, 2 Scott, M., 357, 383 Scott, S., 30 Seamons, J., x

Sebah, P., 12, 16, 38 Selberg sieve, 63 Selfridge conjecture, 82

Selfridge, J., 25, 29, 82, 136, 168, 175, 186

Selfridge, J. (with Bateman et al.), 25

593

Selfridge, J. (with Brillhart et al.), 299

Selfridge–Hurwitz residues, 29 Selkirk, A. (with Escott et al.), 234 Semaev, I., 392

Seroussi, G., 319, 357, 375 seven

-th heaven, 430 7-Up, 430

ages of man, 430

as peculiar base, 162 deadly sins, 430 dwarves, 430

league boots, 430 magic of, 430

pillars of wisdom, 430 seas, 430

wonders, 430

SHA-1 hash function, 390, 391, 394 Shafer, M., 23

Shallit, J., x, 42, 43, 65, 69, 80, 229, 374

Shamir, A., 389, 424

Shamir, A. (with Rivest et al.), 389 Shanks, D., 7, 80, 236, 246, 248, 250, 347, 349, 359

Shanks–Mestre method, 347, 350, 351, 356, 378, 382, 383

Shlesinger, M., 428 Shnirel’man constant, 20 Shnirel’man, G., 20, 62, 63 Shokrollahi relation, 493 Shokrollahi, M., x, 493, 532

Shokrollahi, M. (with B¨urgisser et al.), 466

Shor algorithm (for quantum factoring), 422, 423, 435

Shor, P., 421–424, 441

Shoup (polynomial) method, 383, 533

Shoup, V., 113, 533

Shparlinski, I. (with Friedlander et al.), 401

Shub, M. (with Blum et al.), 401 Siegel, C., 207, 247, 346 Siegel–Walfisz theorem, 43